In this paper, we have described how subsample and 2-sample IV methods can be used to increase the feasibility and cost-efficiency of MR studies. Our primary conclusion is that for epidemiologic studies with available genetic data and outcome data, MR investigations can be conducted by generating exposure data for a limited representative sample of the study population with very little loss of power as compared with a study with exposure data for all participants. For example, in our simulated data set of 10,000 participants, a realistic sample size for large-scale genetic association studies, obtaining exposure data for approximately 20% of the full sample achieves maximum power when the first-stage R2 is greater than 0.015. This finding is of critical relevance for causal evaluations of exposures that are expensive to measure or impossible to obtain for the full set of participants due to lack of prospectively collected or adequately preserved samples. For IVs with weaker effects on the exposure of interest (R2 < 0.015), a larger subsample with exposure data may be required. Additional analytical information clarifying the relationships among nY, nX, R2, and power is provided in the Web Appendix.
